American Institute of Mathematical Sciences

Advanced
September  2021, 26(9): 4645-4661. doi: 10.3934/dcdsb.2020306

Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems

Everaldo de Mello Bonotto 1,, , Matheus Cheque Bortolan 2, , Rodolfo Collegari 3, and  José Manuel Uzal 4,

1. 

Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos - SP, 13566-590, Brazil
2. 

Departamento de Matemática, Centro de Ciências Físicas e Matemáticas, Universidade Federal de Santa Catarina, Florianópolis-SC, 88040-900, Brazil
3. 

Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia-MG, 38400-902, Brazil
4. 

Departamento de Estatística, Análise Matemática e Optimización & Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

* Corresponding author

Received  April 2020 Revised  August 2020 Published  September 2021 Early access  October 2020

Fund Project: The first author is partially supported by FAPESP grant 2016/24711-1 and CNPq grant 310497/2016-7. The second author is partially supported by CNPq, project # 407635/2016-5. The third author is partially supported by FAPEMIG, project # APQ-00371-18.The fourth author is partially supported by the predoctoral contact BES-2017-082334

In this paper we investigate the long time behavior of a nonautonomous dynamical system (cocycle) when its driving semigroup is subjected to impulses. We provide conditions to ensure the existence of global attractors for the associated impulsive skew-product semigroups, uniform attractors for the coupled impulsive cocycle and pullback attractors for the associated evolution processes. Finally, we illustrate the theory with an application to cascade systems.

Keywords: Impulses, global attractor, pullback attractor, skew-product semiflow, cascade systems.
Mathematics Subject Classification: Primary:35B41;Secondary:34A37, 35R12.
Citation: Everaldo de Mello Bonotto, Matheus Cheque Bortolan, Rodolfo Collegari, José Manuel Uzal. Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4645-4661. doi: 10.3934/dcdsb.2020306
References:
[1]

N. U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control Optim., 42 (2003), 669-685.  doi: 10.1137/S0363012901391299.

[2]

M. Benchora, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006. doi: 10.1155/9789775945501.

[3]

E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations, 262 (2017), 3524-3550.  doi: 10.1016/j.jde.2016.11.036.

[4]

E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Methods in the Appl. Sci., 40 (2017), 1095-1113.  doi: 10.1002/mma.4038.

[5]

E. M. Bonotto and P. Kalita, On attractors of generalized semiflows with impulses, The Journal of Geometric Analysis, 30 (2020), 1412-1449.  doi: 10.1007/s12220-019-00143-0.

[6]

E. M. Bonotto, Flows of characteristic $0^+$ in impulsive semidynamical systems, J. Math. Anal. Appl., 332 (2007), 81-96.  doi: 10.1016/j.jmaa.2006.09.076.

[7]

E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Impulsive surfaces on dynamical systems, Acta Mathematica Hungarica, 150 (2016), 209-216.  doi: 10.1007/s10474-016-0631-0.

[8]

E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, J. Differential Equations, (2015), 2602-2625. doi: 10.1016/j.jde.2015.03.033.

[9]

M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522.  doi: 10.1016/j.jde.2014.04.008.

[10]

M. C. Bortolan and J. M. Uzal, Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions, Discrete Contin. Dyn. Syst., 40 (2020), 2791-2826.  doi: 10.3934/dcds.2020150.

[11]

B. Bouchard, N.-M. Dang and C.-A. Lehalle, Optimal control of trading algorithms: A general impulse control approach, SIAM J. Finan. Math., 2 (2011), 404-438.  doi: 10.1137/090777293.

[12]

T. Cardinali and R. Servadei, Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Am. Math. Soc., 132 (2004), 2339-2349.  doi: 10.1090/S0002-9939-04-07343-5.

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[14]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/coll/049.

[15]

S. Dashkovskiy, O. Kapustyan and I. Romaniuk, Global attractors of impulsive parabolic inclusions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1875-1886.  doi: 10.3934/dcdsb.2017111.

[16]

M. H. A. Davis, X. Guo and G. Wu, Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276-5293.  doi: 10.1137/090780419.

[17]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008. doi: 10.1007/978-1-84628-708-4.

[18]

J. A. Feroe, Existence and stability of multiple impulse solutions of a nerve equation, SIAM J. Appl. Math., 42 (1982), 235-246.  doi: 10.1137/0142017.

[19]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications, 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[20]

W. M. Haddad and Q. Hui, Energy dissipating hybrid control for impulsive dynamical systems, Nonlinear Anal., 69 (2008), 3232-3248.  doi: 10.1016/j.na.2005.10.052.

[21]

S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.  doi: 10.1016/0022-247X(90)90199-P.

[22]

K. Li, C. Ding, F. Wang and J. Hu, Limit set maps in impulsive semidynamical systems, J. Dyn. Control. Syst., 20 (2014), 47-58.  doi: 10.1007/s10883-013-9204-5.

[23]

V. F. Rožko, A certain class of almost periodic motions in systems with pulses, Differencial' nye Uravnenja, 8 (1972), 2012-2022. 

[24]

V. F. Rožko, Ljapunov stability in discontinuous dynamical systems, Differencial'nye Uravnenja, 11 (1975), 1005-1012, 1148.

[25]

V. F. Rožko, The almost recurrent and recurrent motions of discontinuous dynamical systems, Differencial'nye Uravnenja, 9 (1973), 1826-1830, 1925.

[26]

H. Song and H. Wu, Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215.  doi: 10.1016/j.jmaa.2006.02.041.

show all references

References:
[1]

N. U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control Optim., 42 (2003), 669-685.  doi: 10.1137/S0363012901391299.

[2]

M. Benchora, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006. doi: 10.1155/9789775945501.

[3]

E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations, 262 (2017), 3524-3550.  doi: 10.1016/j.jde.2016.11.036.

[4]

E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Methods in the Appl. Sci., 40 (2017), 1095-1113.  doi: 10.1002/mma.4038.

[5]

E. M. Bonotto and P. Kalita, On attractors of generalized semiflows with impulses, The Journal of Geometric Analysis, 30 (2020), 1412-1449.  doi: 10.1007/s12220-019-00143-0.

[6]

E. M. Bonotto, Flows of characteristic $0^+$ in impulsive semidynamical systems, J. Math. Anal. Appl., 332 (2007), 81-96.  doi: 10.1016/j.jmaa.2006.09.076.

[7]

E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Impulsive surfaces on dynamical systems, Acta Mathematica Hungarica, 150 (2016), 209-216.  doi: 10.1007/s10474-016-0631-0.

[8]

E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, J. Differential Equations, (2015), 2602-2625. doi: 10.1016/j.jde.2015.03.033.

[9]

M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522.  doi: 10.1016/j.jde.2014.04.008.

[10]

M. C. Bortolan and J. M. Uzal, Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions, Discrete Contin. Dyn. Syst., 40 (2020), 2791-2826.  doi: 10.3934/dcds.2020150.

[11]

B. Bouchard, N.-M. Dang and C.-A. Lehalle, Optimal control of trading algorithms: A general impulse control approach, SIAM J. Finan. Math., 2 (2011), 404-438.  doi: 10.1137/090777293.

[12]

T. Cardinali and R. Servadei, Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Am. Math. Soc., 132 (2004), 2339-2349.  doi: 10.1090/S0002-9939-04-07343-5.

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[14]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/coll/049.

[15]

S. Dashkovskiy, O. Kapustyan and I. Romaniuk, Global attractors of impulsive parabolic inclusions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1875-1886.  doi: 10.3934/dcdsb.2017111.

[16]

M. H. A. Davis, X. Guo and G. Wu, Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276-5293.  doi: 10.1137/090780419.

[17]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008. doi: 10.1007/978-1-84628-708-4.

[18]

J. A. Feroe, Existence and stability of multiple impulse solutions of a nerve equation, SIAM J. Appl. Math., 42 (1982), 235-246.  doi: 10.1137/0142017.

[19]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications, 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[20]

W. M. Haddad and Q. Hui, Energy dissipating hybrid control for impulsive dynamical systems, Nonlinear Anal., 69 (2008), 3232-3248.  doi: 10.1016/j.na.2005.10.052.

[21]

S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.  doi: 10.1016/0022-247X(90)90199-P.

[22]

K. Li, C. Ding, F. Wang and J. Hu, Limit set maps in impulsive semidynamical systems, J. Dyn. Control. Syst., 20 (2014), 47-58.  doi: 10.1007/s10883-013-9204-5.

[23]

V. F. Rožko, A certain class of almost periodic motions in systems with pulses, Differencial' nye Uravnenja, 8 (1972), 2012-2022. 

[24]

V. F. Rožko, Ljapunov stability in discontinuous dynamical systems, Differencial'nye Uravnenja, 11 (1975), 1005-1012, 1148.

[25]

V. F. Rožko, The almost recurrent and recurrent motions of discontinuous dynamical systems, Differencial'nye Uravnenja, 9 (1973), 1826-1830, 1925.

[26]

H. Song and H. Wu, Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215.  doi: 10.1016/j.jmaa.2006.02.041.

[1]

Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261
[2]

P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883
[3]

Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081
[4]

Juan A. Calzada, Rafael Obaya, Ana M. Sanz. Continuous separation for monotone skew-product semiflows: From theoretical to numerical results. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 915-944. doi: 10.3934/dcdsb.2015.20.915
[5]

Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291
[6]

Bogdan Sasu, A. L. Sasu. Input-output conditions for the asymptotic behavior of linear skew-product flows and applications. Communications on Pure and Applied Analysis, 2006, 5 (3) : 551-569. doi: 10.3934/cpaa.2006.5.551
[7]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[8]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[9]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[10]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[11]

Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060
[12]

Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036
[13]

Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2699-2722. doi: 10.3934/dcdsb.2020028
[14]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[15]

B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077
[16]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215
[17]

I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635
[18]

Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216
[19]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[20]

Yuncheng You. Global attractor of the Gray-Scott equations. Communications on Pure and Applied Analysis, 2008, 7 (4) : 947-970. doi: 10.3934/cpaa.2008.7.947

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (218)
  • HTML views (276)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy